Requirements for gate dielectric thickness for advanced semiconductor devices can lead to significant tunneling. Thus high-dielectric constant (high k) materials, e.g., materials having a dielectric constant that is higher than that of the silicon dioxide, have been recently used. A challenge for the high k dielectric is to minimize the leakage current, e.g., to achieve a leakage current similar to that of silicon dioxide.
The leakage via dielectric layers can come from two main sources: tunneling of the electrons with energies in the range between the two Fermi energies of the two electrodes (the two Fermi energies having been separated by the finite voltage across the capacitor), and the propagation of thermionic excitations.
The electronic thermion excitations propagate easily (often nearly ballistically) if their energies exceed the electron Schottky barrier, approximately equal to the conduction band offset bCBM of the dielectric relative to the unbiased Fermi level. There also exist thermionic hole excitations that propagate if their energies are below the hole Schottky barrier, approximately equal to the valence band offset bVBM relative to the unbiased Fermi level. The thermionic currents are exponentially small in the value of the barriers bCBM and bVBM, e.g., proportionally to exp(bx/kT) with bx being bCBM or bVBM. The values of bCBM and bVBM depend on material properties, such as the affinity and band gap of the dielectric material and the effective work function of the electrode, but typically bCBM<bVBM. Thus, electronic excitations are typically the main source of the thermionic leakage. The tunneling currents can have a different exponential dependence on the band gap value, such as,
      ⅇ          d      ⁢                        kb          CBM                      or      ⅇ          d      ⁢                        kb          VBM                    in case of tunneling under flat-band conditions, where d is the material thickness and k is a numerical constant. Other mechanisms exhibiting more complex exponential dependence, e.g. those involving both tunneling and electron-phonon scattering (such as trap-assisted tunneling mechanism), can also contribute to leakage, for example in materials containing defects or impurities.
Reduction of tunneling leakage can be achieved by using a high-k material of increased equivalent thickness d, e.g., using materials having very high dielectric constant values. However, very high k strongly correlates with low value of the dielectric band gap, bCBM−bVBM. On the other hand, thermionic leakage becomes dominant if one or both barriers bCBm and bVBm are low, and/or the temperature T is high, and/or the dielectric thickness d is large. Thus, attempts to decrease tunneling often lead to an increase in thermionic leakage.